Lesson 29: Probability of Independent Events

1. Lesson 29: Probability of Independent Events E.3.1.1 Find the theoretical probability of a simple and/or compound event. E.3.1.2 Find the theoretical probability of an event not occurring.

2. Vocabulary • Event- Something that happens • Probability (P)- likelihood that the event will happen. • Independent- stands alone. Does not rely on another event occurring.

3. Example 1: • What is the probability that the spinner will stop at a section marked B? • Step 1: Find the total number of possible outcomes. • The possible number of outcomes is the number of equal sections on the spinner. There are 8 possible outcomes. • Step 2: Find the number of favorable outcomes. • The number of favorable outcomes is the number of sections marked B. There are 3 favorable outcomes.

4. Example 1 (Cont.) • Step 3: Use the probability ratio. • P = number of favorable outcomes = 3 possible number of outcomes 8 • Solution: The probability of the spinner stopping on B is 3 . 8

5. Hints: • If it is impossible for an event to happen, then its probability is 0. • If it is certain that an event will happen, then its probability is 1. • It is certain that a given event will happen or not. Therefore, the sum of the probability of an event happening and the probability of that event not happening is 1. • (P)- Probability of an event happening. • (1 – P)- Probability of an event not happening.

6. Example 2: • What is the probability of the spinner stopping on a letter that is not B? • Use the fact that the probability of an event event not happening is 1 – P. • Step 1: What is the probability of the spinner stopping on B? P = # of favorable outcomes = 3 possible # of outcomes 8

7. Example 2 (Cont.) • Step 2: What is the probability of the spinner not stopping on B? 1 – P = 1 - 3 = 5 8 8 • Solution: The probability of the spinner stopping on a letter that is not B is 5 . 8

8. Example 3: • What is the probability of spinning a 2 on this spinner and rolling a number greater than 2 on this number cube with faces numbered 1 to 6?

9. Example 3 (Cont.) • Strategy: Find the probability of each independent event and then find the product of the probabilities. • Step 1: Find the probability of spinning a 2 on the spinner. There is 1 favorable (spinning a 2) and 3 possible outcomes. • P= # of favorable outcomes = 1 3

10. Example 3(Cont.) • Step 2: Find the probability of rolling a number greater than 2. • There are 4 favorable outcomes (rolling a 3, 4, 5, or 6), and 6 possible outcomes. • P = number of favorable outcomes = 4 = 2 possible number of outcomes 6 3 • Step 3: Find the product of the probabilities. • 1 * 2 = 2 3 3 9 • Solution: The probability of spinning a 2 and rolling a number greater than 2 is 2 . 9

11. Check It Out with the COACH. • What is the probability of spinning an odd number and rolling an even number? • On the spinner, the total number of outcomes is _____________. • The number of favorable outcomes for spinning an odd number is __________________. • So the probability of spinning an odd number is _______________. • On the number cube, the total number of outcomes is _________________. • The number of favorable outcomes for rolling an even number is __________________.

12. Check It Out with the COACH. • So the probability of rolling an even number is ______________________. • In simplest form, the probability of rolling an even number is ___________________. • Multiply the probabilities of the independent events and, if possible, simplify the result: • _________________ * _______________= ________ • The probability of spinning an odd number and rolling an even number is ___________________.

13. Sample Test Questions: • Use the spinner for Question 1 – 5. • 1. What is the probability of spinning a 12? • A. 1 16 • B. 1 12 • C. 1 8 • D. 1 4

14. Answer 1: • The correct answer is C. 1 . 8 • There are 8 possible outcomes. • There is 1 favorable outcome. • P = # of favorable outcomes total # of outcomes

15. Question 2: • 2. What is the probability of spinning a number less than 7? • A. 1 2 • B. 3 8 • C. 1 4 • D. 1 8

16. Answer 2: • The correct answer is B. 3 . 8 • There are 8 total outcomes. • There are 3 numbers less than 7.

17. Question 3: • What is the probability of spinning an even number? • A. 0 • B. 1 4 • C. 1 8 • D. 1

18. Answer 3: • The correct answer is D. 1 • It is certain this event will happen. • When it is certain that event will happen, the probability is 1.

19. Question 4: • What is the probability of spinning a number that is greater than 16? • A. 0 • B. 1 4 • C. 1 8 • D. 1

20. Answer 4: • The correct answer is A. 0 • It is impossible for this event to occur. • Therefore, the probability is 0.

21. Question 5: • 5. What is the probability of spinning a number that is not 8? • A. 1 8 • B. 1 4 • C. 7 8 • D. 1

22. Answer 5: • The correct answer is C. 7 8 • There are 8 possible outcomes. • There are 7 favorable outcomes.

23. Question 6: • 6. What is the probability of rolling a 5 and getting tails? • A. 1 12 • B. 1 8 • C. 1 6 • D. 1 2

24. Answer 6: • The correct answer is A. 1 12 • The probability of rolling a 5 is: 1 6 • The probability of getting tails is: 1 2 • Therefore, 1 * 1 = 1 6 2 12

25. Question 7: • What is the probability of rolling a prime number and getting tails? • A. 1 6 • B. 1 4 • C. 1 2 • D. 3 4

26. Answer 7: • The correct answer is : B. 1 4 • The probability of a prime number is 3 6 • The probability of tails is 1 2 • Therefore, 3 * 1 = 3 = 1 6 2 12 4

27. Question 8: • 8. What is the probability of spinning a number greater than 2 an rolling a number greater than 2? • A. 1 48 • B. 1 14 • C. 1 4 • D. 1 2

28. Answer 8: • The correct answer is D. 1 2 • The probability of spinning a number greater than 2 is 6 8 • The probability of rolling a number greater than 2 is 4 6 • Therefore, 6 * 4 = 24 = 1 8 6 48 2

29. Question 9: • 9. What is the probability of spinning a prime number and rolling a prime number? • A. 1 8 • B. 1 4 • C. 1 2 • D. 1

30. Answer 9: • The correct answer is B. 1 4 • The probability of spinning a prime # is 4 8 • The probability of rolling a prime # is 3 6 • Therefore, 4 * 3 = 12 = 1 8 6 48 4

31. Question 10: • 10. What is the probability of spinning a number that is not 5 and rolling a number that is not 5? • A. 1 48 • B. 35 48 • C. 5 6 • D. 7 8

32. Answer 10: • 10. The correct answer is B. 35 48 • The probability of not spinning a 5 is 7 8 • The probability of not rolling a 5 is 5 6 • Therefore, 7 * 5 = 35 8 6 48

33. Open-Ended Question 11: • 11A. A bag contains 2 red marbles, 3 blue marbles, and 5 green marbles. If you reach into the bag without looking, what is the probability of drawing a red marble? (Simplest Form)

34. Answer 11A: • The probability of drawing a red marble is 1 5. • There are 2 red marbles. • There are a total of 10 marbles.

35. 11B: Open-Ended • 11B: You reach into the bag and draw one marble without looking. You put it back then you draw another marble without looking. What is the probability that both marbles are blue? (Simplest Form)

36. Answer 11B: • The probability is 9 100 • The probability of the first draw is 3 10 • The probability of the second draw is 3 10 • Therefore, 3 * 3 = 9 10 10 100